Q.The relation between the height of the plant ($y$ cm) with respect to exposure to sunlight is governed by the equation $y = 4x - \frac{1}{2}x^2$, where $x$ is the number of days exposed to sunlight.

The key idea is differential calculus, used to find the rate of change and optimize a function.

1. The rate of growth of the plant with respect to sunlight is given by the derivative of the height function $y$ with respect to the number of days $x$.

   $\frac{dy}{dx} = \frac{d}{dx} \left( 4x - \frac{1}{2}x^2 \right) = 4 - x$

2. To find the maximum height, we set the rate of growth to zero.

   $4 - x = 0 \implies x = 4$

   Substituting $x=4$ days into the height equation: …

The problem asks us to analyze the growth of a plant, described by a quadratic equation, in two parts: first, to find its rate of growth, and second, to determine when it reaches its maximum height and what that height is.

The height of the plant $y$ is given by the equation $y = 4x - \frac{1}{2}x^2$, where $x$ is the number of days exposed to sunlight.

Concept and Intuition

1. Rate of Growth: In mathematics, the "rate of change" of a quantity with respect to another is found by taking the derivative of the first quantity with respect to the second. Here, we need the rate of growth of the plant (change in height, $y$) with respect to sunlight exposure (change in days, $x$). This means we need to calculate $\frac{dy}{dx}$.

2. Maximum Height: For a continuous and differentiable function, a local maximum or minimum occurs at points where the derivative is zero. Geometrically, this means the tangent line to the curve at that point is horizontal. For a quadratic function of the form $ax^2 + bx + c$, if $a < 0$, the parabola opens downwards, and the vertex represents the global maximum. Setting the first derivative to zero will give us the $x$-value (number of days) at which this maximum occurs. Once we have this $x$-value, we can substitute it back into the original height equation to find the maximum height $y$.

Let's proceed with the calculations.

1. Find the rate of growth of the plant with respect to sunlight. The height of the plant is given by $y = 4x - \frac{1}{2}x^2$. To find the rate of growth, we differentiate $y$ with respect to $x$:

Using the power rule for differentiation ($\frac{d}{dx}(x^n) = nx^{n-1}$) and the linearity of differentiation:

This expression, $4-x$, represents the instantaneous rate of growth of the plant in cm per day.

2. Determine in how many days the plant will attain its maximum height.

A plant attains its maximum height when its rate of growth becomes zero. This is because, at the peak of its growth, the height is momentarily neither increasing nor decreasing.

So, we set the rate of growth $\frac{dy}{dx}$ to zero:

Solving for $x$:

Thus, the plant will attain its maximum height after $4$ days of exposure to sunlight. …